Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

218 MATHEMATICAL PHILOSOPHY
contraction toward 0; if k be negative, the effect is such
an expansion or contraction, followed or preceded by
reflection in O as in a mirror; distances are clearly not
preserved, but distance ratios are; that is, if A, B, C
be any three points and if their respective transforms
under some homothetic transformation be A', B', C', it
is evident that AB/BC=A'B'/B'C'; accordingly, if F'
be the transform of a figure F, the figures are similar,they have the same shape but not the same size,-they
are not congruent: similarity is, then, invariant under all
homothetic transformations, and hence under combinations of them with one another and with displacements;
the displacements and the homothetic transformations
together with all such combinations constitute a group
called the group of similitude transformations; it contains
all and only such space transformations as leave similarity
unchanged. Here, then, is our group definition of shape
geometry: namely, the geometry of shape is the study of
that property of figures which is common to every figure
and its transforms under each and all transformations of the
similitude group. Observe that this definition, unlike
the former one, employs neither the notion of shape in
general, nor that of the shape of a given figure; it employs
only the notion of similarity-sameness of shape.
We ought, I think, to consider one more example of
how a group of transformations serves to determine the
nature and limits of a doctrine and thereby to discriminate
the doctrine from all others. I will again take a geometric
example, but for the sake of simplicity I will choose it
from the geometry of the plane (instead of space). Before
presenting it, let me adduce a yet simpler example of the
same kind taken from the geometry of points in a straight
lne. In Lecture IV, I explained what is meant by a